In Rotman he defined the Augmented singular complex by extending the singular chain complex of a space $...\rightarrow S_2(X) \rightarrow S_1(X) \rightarrow S_0(X) \rightarrow 0$ By defining $ \epsilon (\sum m_x x)= \sum m_x[\emptyset]$.
In spanner he says this map $\epsilon$ must be subjective hence, possibility of considering $X=\emptyset$ goes away.
Hatcher says, we should choose $X$ to be nonempty to avoid getting nonzero homology groups of negative degree.
But Rotman never mentions anything about emptiness of $X$.
Later he gives a problem which says,
If $A\subset X$ , then there is an exact sequence $…\rightarrow \tilde H_n(A)\rightarrow \tilde H_n(X) \rightarrow H_n(X,A)\rightarrow …$ , which ends at
$…\rightarrow \tilde H_0(A) \rightarrow \tilde H_0(X) \rightarrow H_0(X,A) \rightarrow 0$
For, $A\neq \emptyset$ the problem is plain coming from the equality of chain complexes , $\tilde S_*(X)/\tilde S_*(A) = S_*(X)/S_*(A)$
If I put $A=\emptyset$ then depending upon $X$ to be non empty or empty various cases are coming and some of them are contradictory.
For example, if $A=\emptyset$ and $X\neq \emptyset$ then from the exact sequence $\tilde H_0(A)=0, H_0(X,A)= H_0(X)$ so, $\tilde H_0(X)\cong H_0(X)$, but this contradicts the relation of 0th reduced homology and 0th homology group in terms of rank, when X has finitely many path components.
Is there any way to deal with this?
Does the usual practice of reduced homology groups deal with nonempty spaces only?